Instructions
Go to Goldenrod City and obtain the "Silver Wing" from the Radio Director. The "Silver Wing" is an item that you need to store in order to get Lugia.
Go to the Whirl Island, which is located between Cianwook City and Olivine City.
Go through the dungeon once you arrive to the island. Be sure to have Whirlpool and Surf to use with one of your pokémon.
Go to the very bottom of the dungeon. Surf all the way to Lugia, who will be waiting for you.
Battle Lugia to get its health down so that you can capture it. Make sure that your pokémon are at least near level 40 before battling Lugia. Also, make sure you have full restores and ultra balls. Weaken Lugia enough to make him fall asleep and then throw balls at him until you finally capture him.